Semigroup Forum

, Volume 86, Issue 1, pp 192–201 | Cite as

On explicit representation and approximations of Dirichlet-to-Neumann semigroup

RESEARCH ARTICLE

Abstract

In his book (Functional Analysis, Wiley, New York, 2002), P. Lax constructs an explicit representation of the Dirichlet-to-Neumann semigroup, when the matrix of electrical conductivity is the identity matrix and the domain of the problem in question is the unit ball in ℝ n . We investigate some representations of Dirichlet-to-Neumann semigroup for a bounded domain. We show that such a nice explicit representation as in Lax book, is not possible for any domain except Euclidean balls. It is interesting that the treatment in dimension 2 is completely different than other dimensions. Finally, we present a natural and probably the simplest numerical scheme to calculate this semigroup in full generality by using Chernoff’s theorem.

Keywords

Dirichlet-to-Neumann operator Lax’s Dirichlet-to-Neumann semigroup γ-harmonic lifting 

Notes

Acknowledgements

We wish to thank Professor Ralph deLaubenfels who was the instigator of this method, for his collaboration with the first author which ends up with this paper.

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Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.School of MathematicsInstitute for Research in Fundamental Sciences (IPM)TehranIran
  2. 2.Laboratoire de MathématiquesUniversité de PoitiersChassneuil du Poitou CedexFrance

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